Emergence of digit-like structures through mechanical instabilities:A hybrid computational and experimental studySteffen PlunderSeirin-GASHBi Fusion Grant with Rio Tsutsumi & Antoine DiezASHBi Retreat 2025
Big PictureRegenerative medicine: "replacing, engineering or regenerating cells, tissues or organs to restore or establish normal function"This project: study of growth of limb bud and digit formation → tissue with a shapeSullivan, A. E. & Santos, S. D. The ever-growing world of gastruloids: autogenous models of mammalian embryogenesis. Current Opinion in Genetics & Development (2023).
Morphology: How Does Nature Create Shapes?How do cells organize themselves into the requires shapes?Morishita, Y. et al. An archetype and scaling of developmental tissue dynamics across species. Nat Commun (2023).
Morphology: How Does Nature Create Shapes?Is everything determined by morphogens?Gilbert, S. F. & Barresi, M. J. F. Developmental Biology. (2018).AER(apicalectodermal ridge)How are complex shapes like digits established?
Complex Shapes & InstabilitiesOften symmetric states are stable...However, if they are instable, this can breaksymmetry and create complex shapes.There are many types of instabilities in nature:ResonanceFluid interface (Saffman–Taylor instability)Ma, X., Zhong, M., He, Y., Liu, Z. & Li, Z. Fingering instability in Marangoni spreading on a deep layer of polymer solution. Physics of Fluids (2020).Turing instability (diffusion based) Walker, B. J., Townsend, A. K., Chudasama, A. K. & Krause, A. L. VisualPDE: Rapid Interactive Simulations of Partial Differential Equations. Bull Math Biol (2023).
Current state: We know many details about the signals around digit formation! Raspopovic, J., Marcon, L., Russo, L. & Sharpe, J. Digit patterning is controlled by a Bmp-Sox9-Wnt Turing network modulated by morphogen gradients. Science (2014). Required model inputs:Fgf signal from the AERTuring model (with pattern via diffusion based instability)Current gap in existing models:How do signals and tissue growth interact?
Our ApproachLimb Budin vivoTuring Modelinput: shape growth + signalsoutput: signal patternAgent-Based Modelsinput: signals + patternoutput: shape growthcomplexitymath → biologyDigit Organoidin vitrodigit formationvia in vitro modelgives access to more data!reduced number ofmathematical assumptionsOur Hybrid Model input: experimental conditionoutput: shape growth + pattern + signalsData Analysis3D tracking dataPropose MechanismKey finding from modelling:Formation of digit-like structures in organoidcould be explained by an interplay ofa new mechanical instability (not Turing) and self-organized Wnt gradient.
Construction of a Mathematical Model for Digit Organoids
Digit OrganoidRio TsutsumiDistal CellsProximal CellsmixSerumFgf8+Wnt5a/Serum9000 cells Fgf8+Wnt5a/Serum27000 cellsfull in vivo limb bud has ~200.000 cells Note: No signaling center & no AER!
Question: How can proximal & distal cells self-organizei) sorting ii) symmetry breaking & iii) elongation/digit-formation?Fgf+Wnt/SerumWe use mathematical modelling to test how this could be explained.
How do cells move?Analysis of cell tracking data
Core modelling question: How can an organoid elongate?Micromass experiments: Evidence for chemotaxis distanceof attraction~ 300µmIndicates potential role ofmorphogens.
Candidate elongation mechanism: Planar Cell Polarity (PCP)Vangl2 polarization shows potential PCPDirection of extensionPCP leads to convergent-extensionDestabilisation of adhesiveinteraction at Vangl/Fzd interfacesWnt can induce PCP:Well-known role of Fgf/Wnt gradient in limb bud:In vivo evidence of Vangl2 in limb bud:Proxiaml - Distal axistodo: add references
The challenges of measuring converent-extension in vivoAccumulation effect:timevelocitylarge velocity at the tipis misleading!Related dataMorishita, Y. et al. An archetype and scaling of developmental tissue dynamics across species. Nat Commun (2023).(ex-plants)PIVParada, C. et al. Mechanical feedback defines organizing centers to drive digit emergence. Developmental Cell (2022).
The Green strain tensor: A way to measure deformation without bias & accumulation effects!Pipeline:0. raw 3D tracks2. interpolate velocity field(Gaussian kernel smoothening)1. cleanup 3. compute deformation mapfollowing interpolated velocities4. compute at each point the Green strain tensorframe 1frame 30 deformation map Main idea: Track how local sphere would deform within the tissue(+ tiny amount of math to make it less sensitive to noise/rotations)
Convergent-Extension IndexElongation is drive by convergent-extensionGreen strain tensorLocalization ofconvergent extensionat distal-proximal interface
Construction of a mathematicalmodel
Related works: Mathematical models for elongating organoidsOkuda, S. & Sato, K. Polarized interfacial tension induces collective migration of cells, as a cluster, in a 3D tissue. Biophysical Journal (2022). Vertex-basedSphere-basedOkuda, S., Miura, T., Inoue, Y., Adachi, T. & Eiraku, M. Combining Turing and 3D vertex models reproduces autonomous multicellular morphogenesis with undulation, tubulation, and branching. Sci Rep (2018). Germann, P., Marin-Riera, M. & Sharpe, J. ya||a: GPU-Powered Spheroid Models for Mesenchyme and Epithelium. cels (2019). Up to our knowledge:No hybrid (signal/mechanical) model for elongating organoid of mesenchymal cellsor limb buds.
Initial agent-based organoid modelWe describe each cell by a - position (in 3D space)- radius (spherical cell model)- celltype (proximal or distal)N = 10.000 ~ 30.000Initial position & cell type are random.Rules Repulsion DifferentialAdhesionExperimental dataanglecortical tensionadhesive forcestronger adhesion force →Newton's 3rd LawWith structurally stable ECMForce balancecell forwardpulling on ECMBasic Physics: How can cells move forward?Organoid setting (no structurally stable ECM in matrigel)#1#2#3pulled back by #2pulling on #3pulling on #2pulled back by #1 proximal/distal✔ sorting✘ sym break✘ elongation
Optional math:All three cases are included in a simple (symmetric) force function: (direction of adhesion) (signal direction)(aligment of signal & direction of force)Planar Cell Polarity (PCP) between distal cellsWnt signaldistal-distalprimary lateral adhesion Convergent ExtensionWntChemotaxis of distal cellsthrough proximal cellsdistal-proximal Wnt signalchemotaxis via biased adhesion ChemotaxisWntProximal cells ignoremorphogenfrontlateralrearcell-cell adhesion forcesD-DP-PD-PA minimal mathematical model for PCP and more...
GPU-accelerated simulations for realistic cell counts (>10.000 particles) → github.com/antoinediez/SisypheWnt(more complex examples later...)Convergent-Extension
Unexpected consequence of PCP:a new mechanical instability...
Recall: without PCP/Wnt gradient Assume there would be an radial Wnt gradient field...WntIllustrative 2D example: Key modeling insight: PCP leads to an mechanical instability which leads to digit-formation.
Establishing a Wnt gradientvia a simple pathway
How could a Wnt gradient be established?Digit OrganoidFgf+Wnt/SerumKnown:1. Fgf (in cell junctions):inside: short range diffusion + fast decay outside: available en mass2. Wnt (membrane bound):inside: medium range diffusion + slow decay outside: [irrelevant for math model]Fgf8 inaggregatedpERK FGF pathway activityTen Berge, D., Brugmann, S. A., Helms, J. A. & Nusse, R. Wnt and FGF signals interact to coordinate growth with cell fate specification during limb development. Development (2008). 3. Fgf signal triggers distal cells to emit WntHybrid Agent-based + Reaction Diffusion Model
Hybrid model explains possible mechanism fori) sorting ii) symmetry breaking & iii) elongation/digit-formation!
Symmetry breaking via a feedback between cell dynamics and morphogenesSymmetric start:- spherical organoid- radial symmetric signaldistal/proximalFgfWntSymmetry breaking:- Mechanical instability: leads to asymmetric distal cell clusters- feedback with morphogensvia Fgf+distal-Wnt pathwayFgfWntEstablished Wnt gradient:- Distal cells keep emitting Wnt, maintaining distal-proximalWnt gradient.- Proximal cells do nothing, butare essential to create Wnt gradient!FgfWnt
Comparison with experimental data
Simulations recapture behaviour at different ratios of distal cells:Digit Organoid SimulationsRatio of distal cells within organoid(higher total cell count)
Green strain tensorConvergent-Extension Index3D tracking data: Digit Organoid Our hybrid model (in silico)Localization ofconvergent extensionat distal-proximal interface
ConclusionMathematical model suggests:Digit-formation might not reply only on Turin instability (diffusion based)but also use a mechanical instability! Comparing simulations and digit organoid data suggest that PCP + Fgf-Wnt crosstalk creates digit-like elongations.Supporting the idea digit organoid elongations recaptures in vivo mechanism.
Closing remarks from a mathematical point of viewUsing statistical physics, Antoine could derive the underlying equation for densities of distal cells: (much easier to analyze,finding concepts)(good for computers,bad for theorety/insights.)Cahn-Hilliard PDE Open mathematicalquestions to characterize instabilityWork in progress: Antoine Diez & Hyunjoon ParkEquation for fluid interfaces:gasoilYe, Y. & Lin, J. Fingering Instability Accelerates Population Growth of a Proliferating Cell Collective. Phys. Rev. Lett. (2024).Recently also studied for tumor growth:Physicists called this type "fingering" instability.(Up to our knowledge, this type of instability was never been linked to digit/finger growth so far.)
BiMed-Math GroupSungrim Seirin-LeeAntoine DiezTsubasa SukekawaYing XieJinghao ChenTamaki WakamotoMichito UjinoTakahiro HiragaChie ShiraiASHBiSupport and nuturing scienfic exchange:Makoto ShidaFumi KomoriSpyros GoulasAll members of Alev-GTakafumi IchikawaRio TsusumiAntoine DiezThank YouASHBi Fusion Grant
EdU test: Proximal and distal cellsproliferate at a similar rate.Fgf/WntWe assume proliferation has nomajor effect within digit organoid.